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We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.
We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere Sn⊆Rn+1 is maximized by the union of two disjoint, equal, geodesic balls among all subsets of Sn of prescribed volume. In fact, the result holds in a stronger version, involving the harmonic mean of the eigenvalues of order 2 to n, and extends to densities. A (surprising) consequence occurs on the maximality of a geodesic ball for the first nontrivial eigenvalue under the volume constraint: the hemisphere inclusion condition of the Ashbaugh-Benguria result can be relaxed to a weaker one, namely empty intersection with a geodesic ball of the prescribed volume. Although we do not prove that this last inclusion result is sharp, for a mass less than the half of the sphere, we numerically identify a density with higher first eigenvalue than the corresponding geodesic ball and with support equal to the full sphere S2.
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